Least-squares inverse filtering
A well-behaved input wavelet, such as (1, - 12) as opposed to (- 12, 1), has a z-transform whose inverse can be represented by a convergent series. Then the inverse filtering described above yields a good approximation to a zero-lag spike output (1, 0, 0). Can we do even better than that?
Formulate the following problem: Given the input wavelet (1, - 12), find a two-term filter (a, b) such that the error between the actual output and the desired output (1, 0, 0) is minimum in the least-squares sense.
Compute the actual output by convolving the filter (a, b) with the input wavelet (1, - 12) (Table 2-7). The cumulative energy of the error L is defined as the sum of the squares of the differences between the coefficients of the actual and desired outputs:
$ {L=(a-1)^{2}+(b-{\frac {a}{2}})^{2}+(-{\frac {b}{2}})^{2}}. $ ()
The task is to find coefficients (a, b) so that L takes its minimum value. This requires variation of L with respect to the coefficients (a, b) to vanish (Section B.5). By simplifying equation (14), taking the partial derivatives of quantity L with respect to a and b, and setting the results to zero, we get
$ {{\frac {5}{2}}a-b=2}, $ ()
and
$ -a+{{\frac {5}{2}}b=0}. $ ()
We have two equations and two unknowns; namely, the filter coefficients (a, b). The so-called normal set of equations (15a) and (15b) can be put into the following convenient matrix form
$ {\begin{pmatrix}{5}/{2}&-1\\-1&{5}/{2}\end{pmatrix}}{\begin{pmatrix}a\\b\end{pmatrix}}={\begin{pmatrix}2\\0\end{pmatrix}}. $ ()
By solving for the filter coefficients, we obtain (a, b): (0.95, 0.38). Design and application of this least-squares inverse filter are summarized in Table 2-7.
To quantify the spikiness of this result and compare it with the result from the inverse filter in Table 2-3, compute the energy of the errors made in both (Table 2-8). Note that the least-squares filter yields less error when trying to convert the input wavelet (1, - 12) to a spike at zero lag (1, 0, 0).
| Filter Design | ||||||||||||
| Input Wavelet | $ w\left(t\right):\left(1,\ -{\frac {1}{2}}\right) $ | |||||||||||
| The z-Transform | $ W\left(z\right)=1-\ {\frac {1}{2}}z $ | |||||||||||
| The Inverse | $ F\left(z\right)=1\ +\ {\frac {1}{2}}z\ +\ {\frac {1}{4}}z^{2}\ +\ \cdots $ | |||||||||||
| The Inverse Filter | $ f\left(t\right):\left(1,\ {\frac {1}{2}},\ {\frac {1}{4}},\ \cdots \right) $ | |||||||||||
| Filter Application | ||||||||||||
| Truncated Inverse Filter | $ \left(1,\ {\frac {1}{2}}\right) $ | |||||||||||
| Input Wavelet | $ \left(1,\ -{\frac {1}{2}}\right) $ | |||||||||||
| Actual Output | $ \left(1,\ 0,\ -{\frac {1}{4}}\right) $ | |||||||||||
| Desired Output | (1, 0, 0) | |||||||||||
| Convolution Table: | ||||||||||||
| 1 | - 12 | Output | ||||||||||
| 12 | 1 | 1 | ||||||||||
| 12 | 1 | 0 | ||||||||||
| 12 | 1 | - 14 | ||||||||||
We now examine the performance of the least-squares filter with the input wavelet (- 12, 1). Note that the inverse filter produced unstable results for this wavelet (Table 2-5). We want to find a two-term filter (a, b) that, when convolved with the input wavelet (- 12, 1), yields an estimate of the desired spike output (1, 0, 0) (Table 2-9). As before, the least-squares error between the actual output and the desired output should be minimal.
| Filter Design | ||||||||||||
| Input Wavelet | $ w\left(t\right):\left(-{\frac {1}{2}},\ 1\right) $ | |||||||||||
| The z-Transform | $ W\left(z\right)=-{\frac {1}{2}}+z $ | |||||||||||
| The Inverse | $ F\left(z\right)=-2\ -\ 4z\ -\ 8z^{2}\ -\ \cdots $ | |||||||||||
| The Inverse Filter | f(t): (−2, −4, −8, …) | |||||||||||
| Filter Application | ||||||||||||
| Truncated Inverse Filter | (−2, −4) | |||||||||||
| Input Wavelet | $ \left(-{\frac {1}{2}},\ 1\right) $ | |||||||||||
| Actual Output | (1, 0, −4) | |||||||||||
| Desired Output | (1, 0, 0) | |||||||||||
| Convolution Table: | ||||||||||||
| - 12 | 1 | Output | ||||||||||
| −4 | −2 | 1 | ||||||||||
| −4 | −2 | 0 | ||||||||||
| −4 | −2 | −4 | ||||||||||
The cumulative energy of the error is given by
$ {L=(-{\frac {a}{2}}-1)^{2}+(-{\frac {b}{2}}+a)^{2}+b^{2}}. $ ()
| Filter Design | |||||||||
| Convolution of the filter (a, b) with input wavelet (1, - 12): | |||||||||
| 1 | - 12 | Actual Output | Desired Output | ||||||
| b | a | a | 1 | ||||||
| b | a | b − a/2 | 0 | ||||||
| b | a | −b/2 | 0 | ||||||
| Filter Application | |||||||||
| Least-Squares Filter | (0.95, 0.38) | ||||||||
| Input Wavelet | (1, −0.5) | ||||||||
| Actual Output | (0.95, −0.09, −0.19) | ||||||||
| Desired Output | (1, 0, 0) | ||||||||
| Input: $ (1,\ -{\frac {1}{2}}) $ | ||
| Desired Output: (1, 0, 0) | ||
| Actual Output | Error Energy | |
| Inverse Filter | (1, 0, −0.25) | 0.063 |
| Least-Squares Filter | (0.95, −0.09, −0.19) | 0.048 |
| Filter Design | |||||||||
| Convolution of the filter (a, b) with input wavelet $ (-{\frac {1}{2}},\ 1) $: | |||||||||
| - 12 | 1 | Actual Output | Desired Output | ||||||
| b | a | −a/2 | 1 | ||||||
| b | a | −b/2 + a | 0 | ||||||
| b | a | b | 0 | ||||||
| Filter Application | |||||||||
| Least-Squares Filter | (−0.95, −0.19) | ||||||||
| Input Wavelet | (−0.5, 1) | ||||||||
| Actual Output | (0.24, −0.38, −0.19) | ||||||||
| Desired Output | (1, 0, 0) | ||||||||
| Input: $ (-{\frac {1}{2}},\ 1) $ | ||
| Desired Output: (1, 0, 0) | ||
| Actual Output | Error Energy | |
| Inverse Filter | (1, 0, −4) | 16 |
| Least-Squares Filter | (0.24, −0.38, −0.19) | 0.792 |
By simplifying equation (17), taking the partial derivatives of quantity L with respect to a and b, and setting the results to zero, we obtain
$ {\frac {5}{2}}a-b=-1, $ ()
and
$ -a+{\frac {5}{2}}b=0. $ ()
Combine equations (18a,18b) into a matrix form
$ {\begin{pmatrix}5/2&-1\\-1&5/2\end{pmatrix}}{\begin{pmatrix}a\\b\end{pmatrix}}={\begin{pmatrix}-1\\0\end{pmatrix}}. $ ()
By solving for the filter coefficients, we obtain (a, b): (−0.95, −0.19). The design and application of this filter are summarized in Table 2-9.
Table 2-10 shows the results from the inverse filter and least-squares filter quantified. The error made by the least-squares filter is, again, much less than the error made by the truncated inverse filter. However, both filters yield larger errors for input wavelet (- 12, 1) (Table 2-10) as compared to errors for wavelet (1, - 12) (Table 2-8). The reason for this is discussed next.
See also
External links
| find literature about Least-squares inverse filtering |
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